Pseudorandom bits for non-commutative programs

2025-06-02

AI Summary

This paper addresses the pseudorandomness of read-once group products—specifically, group products, block products, and mixed group products—over non-abelian groups, introducing the first unified framework to overcome long-standing barriers posed by composite moduli and non-commutativity. Methodologically, it integrates small-bias distributions, mixing analysis from group representation theory, deterministic noise construction, and block-function reconstruction techniques, establishing the first provable reduction from representation-theoretic mixing to pseudorandomness. It introduces a novel “de-recursified small-bias + deterministic noise” paradigm, supporting arbitrary input orders. Key contributions include: (1) achieving optimal seed length $log(n/varepsilon)$ for $p$-groups; (2) attaining near-optimal seed length for width-$omega$ block products over the quaternion group and arbitrary abelian groups; and (3) constructing the first pseudorandom generator that fools arbitrary modulo-$m$ read-once polynomials—extending prior results limited to $m = 2$.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsSearch and Optimization: Non-convex OptimizationReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Security and Privacy: Applications of cryptographyGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1. We consider read-once group-products over a finite group $G$, i.e., tests of the form $prod_{i=1}^n g_i^{x_i}$ where $g_iin G$, a special case of read-once permutation branching programs. We give generators with optimal seed length $c_G log(n/varepsilon)$ over any $p$-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was $gelog nlog(1/varepsilon)$, even for a fixed order. 2. We give a reduction that"lifts"suitable generators for group products over $G$ to a generator that fools width-$w$ block products, i.e., tests of the form $prod g_i^{f_i}$ where the $f_i$ are arbitrary functions on disjoint blocks of $w$ bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3. Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed $m$ with nearly optimal seed length. Previously this was known only for $m=2$. 4. We give a new generator for products over"mixing groups."The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity -- handling composite moduli -- and points to several classes of groups to be attacked next.

Problem

• Generating pseudorandom bits for non-commutative group programs efficiently
• Lifting generators for group products to fool width-w block products
• Improving seed length for pseudorandom generators in mixing groups

Innovation

• Optimal seed length pseudorandom generators for p-groups
• Lifting generators for group products to block products
• Representation theory-based generators for mixing groups